Dear Family, your Operation Maths guide to Lines and Angles

Dear Family, your Operation Maths guide to Lines and Angles

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Dear Family, given below is a brief guide to understanding the topic of lines and angles as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about lines and angles. The digital resources are organised according to approximate class level:

Understanding Lines and Angles

Line and angles is a strand unit in the Primary Maths Curriculum for 2nd class up. For most people, when they think about angles, they also think about degrees, for example a 90° (ninety degree) angle. Yet in primary school, we don’t introduce degrees, as a way to describe, measure and construct angles, until 5th class. So what are they doing before that?

Initially, children are exploring angles as ‘turns’ i.e. recognising angles in terms of rotation: quarter-turns, half-turns and full turns. The children explore this themselves by turning to show the different turns, in both clockwise (turning right) and anticlockwise (turning left) directions.

In school, the children will also be shown other real-world examples of angles, including angles made by the hands of a clock, by the blades of an open scissors, by a door opening and closing, etc. We also look at angles (or corners/vertices) in 2-D shapes, for example in rectangles and triangles… even the names of these shapes pay homage to the angles that made them what they are today! (triangle = tri (or 3) angle; rectangle = rect (right/proper) angle)

In third class, the children will begin to use the term right angle, as a more mathematically correct way to describe the quarter turn (when movement is involved) or square corner (when there is no movement i.e. the angle is static) that they met in second class. The children will also learn to identify and name a variety of angles: angles less than a right angle (acute angles), angles more than a right angle (obtuse or reflex, more than 2 right angles/a straight angle) and angles equal to 2 right angles (straight angle). Then, in 5th and 6th class, they will begin to use degrees to identify, measure and construct various types of angles. This requires the use of a protractor, from a maths set, and it is not the most obvious or straightforward tool to use, so plenty of practice is required (see video links below in the Digital resources section for 5th and 6th class).

You can’t have an angle without having at least two connecting lines, therefore lines are an integral part of this area of maths. Like angles, not all lines are the same and the children learn to recognise and describe lines as horizontal, vertical, oblique, parallel or perpendicular.

Practical Suggestions for Supporting Children

• Ask your child to teach you about the angles and lines in your home. What different types of lines can be seen? What different types of angles can be seen? Can they name them?
• Line Hunt: ask your child to show you some lines that they can see at home. Ask them to run their finger along the lines so they get a sense of the line’s direction and position.
• Make a right angle finder: From a scrap piece of paper, tear out a large circular shape. Fold the shape in two, and then fold it in two again. The two straight edges/lines meet at a corner to make a right angle. Ask your child:
• ‘Have you heard any other name for this type of angle?’ (square corner, quarter turn)
• ‘Can you find any angles this size in this room? What are they called?’ (right angles)
• ‘Can you find any angles greater than/smaller than right angles in this room?’
• What’s in name? Write out your name in all capital letters. Name the different line types and angle types you can see. Are there any lines of symmetry in the letters? How many lines of symmetry and where? Click here to see some possible answers.
• You gotta hand it to ’em! Look closely at your hand… can you see angles in the lines made by your fingers? Estimate the degrees and then click here to see possible answers.
• Programming If your child does programming, or is interested in trying it out, they could use a free programme such as Scratch to draw various line and angle types.

Digital Resources for Second to Fourth Classes

Turns, Angles and Lines: A series of video lessons from White Rose Maths, including for Year 2,  Describing Turns 1, Describing Turns 2. For Year 3: Right Angles in ShapesCompare Angles, Horizontal & Vertical, Parallel & Perpendicular. For Year 4: Identify Angles and Compare and Order Angles.

Lines & Angles: A series of video lessons from Matholia including What are Angles?Right Angles, Identifying Angles, Parallel Lines and Perpendicular Lines

Khan Academy – Right Angles: Read through and then answer the practice questions. Afterwards, for something more challenging, look at the Fourth Grade Activities, which explore different types of lines and angles. You can also register for a free Khan Academy account to record your progress and explore other topics/grades.

Turns on a compass: Compare the start and end positions of the dial and decide how it turned.

Turn the man: Explore how many times you need to turn the man to match the images.

Right or Left: Which way is the animal facing?

Turtle Diary: Learn about lines and angles and then take the quizzes! Parallel, perpendicular and intersecting lines; Types of Angles 1; Types of Angles 2

Pattern Blocks: Explore the lines and angles that can be made with these interactive pattern blocks. Also includes a protractor feature for measuring the angles.

Geoboard: Make lots of different angles and lines using this interactive geoboard, free from the Math Learning Centre.

I Know It! Classifying Angles (Third Grade) and also in Fourth Grade.

Acute, Obtuse or Right angle: Answer the quiz questions

Math Games: Identify the parallel, perpendicular and intersecting lines

Geometry: a selection of games from ixl.com, including types of angles, obtuse, acute or straight, types of lines. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum.)

Digital Resources for Fifth & Sixth Classes

Angles and Lines: A series of video lessons from White Rose Maths, including Measure with a Protractor, Introduce Angles, Angles in a Triangle

Measuring Angles with a Protractor: Video Tutorial from Two Minute Math

Khan Academy – Measuring Angles: A unit of work exploring angles, including how to understand angles, how to measure angles and decomposing angles. Other relevant lessons include this one on the sum of angles in a triangle and this one on the sum of angles in a quadrilateral. You can also register for a free Khan Academy account to record your progress and explore other areas and/or try more difficult material.

Angle Alien Attack: Defend the Earth from an alien invasion using your knowledge of angles. Choose to read the angles from the protractor or estimate them without a protractor.

Alien Angles: Create a specified angle to destroy the aliens. Challenging, but great for developing the ability to estimate angles. A similar game is Rocket Angles; this time you must estimate and input the measure of the given angle in degrees.

Estimating Angles: In this game you must stop the angle size as near as possible to the target measure in degrees.

Turtle Diary: Learn about lines and angles and then take the quizzes! Parallel, perpendicular and intersecting lines; Types of Angles 1; Types of Angles 2; Angles in Degrees; Estimating Angles

Pattern Blocks: Explore the lines and angles that can be made with these interactive pattern blocks. Also includes a protractor feature for measuring the angles.

Geoboard: Make lots of different angles and lines using this interactive geoboard, free from the Math Learning Centre.

That Quiz – Angles: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. Ensure that the level is set to 1. Each time do the set 10 questions, if you get 10 or 9 correct go up a level, if not stay at that level. Start with only the “Measure” option on the left-hand side ticked, and when you bring the mouse across the screen, it changes into a transparent protractor. Other options included calculating the value of a missing angle in a triangle, (Triangle), and calculating the value of a missing angle in intersecting lines  (Line) or parallel lines (Parallel).

Geometry: a selection of games from ixl.com, including classifying triangles and quadrilaterals. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum.)

Math Games: selection of interactive geometry quizzes, that includes lines and angles.

Dear Family, your Operation Maths guide to 2-D Shapes

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Dear Family, given below is a brief guide to understanding the topic of 2-D shapes, as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about 2-D shapes. The digital resources are organised according to approximate class level:

Understanding 2-D Shapes

Why do children need to learn about shapes? Learning to recognise different shapes not only helps children learn about this maths topic, but, in the early years, it also develops their ability to recognise numbers and letters by their shape. This skill will also transfer to other signs and symbols, be they maths symbols such as +, =, <, > etc., or signs and symbols in the real world e.g. road signs, safety signs etc.

Naming shapes: 2-D shapes is short for two dimensional shapes, i.e. shapes with length and width, but not depth/height. Also called flat shapes, these include circles, squares, rectangles, triangles etc. 2-D shapes can be a bit confusing for both adults and children; for example, a real ball is not a 2-D shape, it is a 3-D object called a sphere, but if a ball is drawn, or shown in a picture, then the flat representation of the ball in the image is now a circle! And a box is not a 2-D shape, it is a 3-D object called a cuboid, but the flat surface of a box is usually the 2-D shape of a rectangle or sometimes a square. So, if looking for 2-D shapes at home, ask the children to examine the flat surface of objects and/or to look at the flat shapes in a picture book or magazine.

Properties of Shapes: 2-D shapes also have properties or characteristics that make them different from other 2-D shapes. A shape with three straight sides and three angles (also referred to as corners or vertices) is always a triangle…but as the children get older they will also realise that some triangles have three equal sides (equilateral), some triangles have only two equal sides (isosceles) and some have no equal sides (scalene). Through an understanding of what makes a shape that shape, the children can start to group shapes with similar properties or characteristics together. So, if exploring 2-D shapes, draw the children’s attention to properties such as the number and type of sides (equal, not equal, straight or curved), the number and type of angles/vertices (equal, not equal, right angles or not).

Children in the senior end of primary school will further classify shapes into named groups, for example, they will identify different types of triangles, polygons (any shape with straight, non-curved sides) and quadrilaterals (four sided shapes, quad = four) and explore the different properties (size/shape of angles, length of sides) that make each one unique. They will learn more detailed terminology about the parts of shapes, especially the parts of a circle. They will also be asked to solve various problems (for example finding the measure of an unknown angle or side) based on what they know already. This is preparing them for the type of geometry they will meet in second-level maths.

Practical Suggestions for Supporting Children

• Shape hunts: Play games like “I spy, with my little eye, something the shape of a rectangle” etc. Again, be careful that you affirm with your child that it is the surface or face of, for example, the door, that is a rectangle, not the entire door (which is in fact another cuboid i.e. a 3-D object). Look out for 2-D shapes on posters, road signs, billboards, wallpaper and in picture books. With older children, encourage them to notice that while each shape group has a key feature in common, each individual shape is different; for example while every triangle must have 3 sides, they also can have different size angles and sides. Look around for different triangles!
• Play, play, play! Encourage your child to play and explore with 2-D shapes as much as possible:
• Make 2-D shapes with sticks, string, playdoh, pastry, creating imprints in sand, mud, pastry, etc. Use construction toys such as Lego, K’nex, Geomag and Plus-Plus to create 2-D shapes and then build them further into 3-D structures.
• Draw 2-D shapes and cut them out, create pictures, patterns, designs etc. Perhaps you have a spirograph toy somewhere in the house? Dig it out and give it a spin (excuse the pun!).
• Solve shape puzzles. One of these are tangram puzzles. This ancient Chinese 7-piece puzzle, provides an excellent way to develop a child’s ability to manipulate and visualise shapes. You can often buy reasonably priced plastic or wooden tangram puzzles in local book and toy shops. You can also print out a set of trangram pieces and use them to solve the numerous puzzles available on line. Or you can play an interactive tangram puzzle game.
• Other very worthwhile shape puzzles include tetrominoes (like the Tetris game of old) and pentominoes. You can also download games, based on many of these shape puzzles to your device; just search your app store for tangrams, tetrominoes/tetris and/or pentominoes.
• Programming If your child does programming, or is interested in trying it out, they could use a free programme such as Scratch to draw various types of 2-D shapes.

Digital Resources for Infants

The Number Jacks have quite a number of 2-D shape-based episodes including Round and Round, Square Dancing and Very Shapely.

Shapes Songs Collection: a collection of songs that teach children the names of common shapes. For more links to shape videos and songs, click here.

Identifying and Naming 2D Shapes:  A video lesson from Matholia identifying common 2D shapes in the environment. Follow this up with 2-D Shapes, a video lesson further exploring common flat (2-D) shapes, including squares, circles, triangles and rectangles.

Happy Numbers Pre-Kindergarten: Pupils could start the activities in Module 2, Topic A, and then progress to the Shape activities in Kindergarten, Module 2 also.

Shape Monsters: an ideal introduction to 2-D shapes for young children. Children need to feed the monsters with the correct shapes. The monsters then say the name of the shape they’ve eaten.

Pattern Blocks: Make numerous designs, pictures etc with these interactive pattern blocks. You can also choose a puzzle to complete.

Geoboard: Make lots of different shapes using this interactive geoboard, free from the Math Learning Centre.

Kid’s Tangrams: a simple version of the puzzle that would suit infants.

Flat Shapes: A selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. Activity K1-K6 are all about flat, 2-D shapes.

Math Games: a whole suit of geometry games, for all class levels; choose the skill you want to practice.

Digital Resources for First and Second Classes

NB: Children in first and second may also enjoy the links for infant classes, above

White Rose Geometry: a series of lessons on 2-D and 3-D shapes. These lessons could be followed up with other geometry lessons in year 2

Khan Academy – Shapes (First Grade): Watch the videos and then answer the practice questions. Afterwards, for something more challenging, look at the Second Grade Activities. You can also register for a free Khan Academy account to record your progress and explore other topics/grades.

Happy Numbers First Grade: Pupils could start the activities in Module 5, Topic A.

Tangrams: interactive puzzle pieces that can be rotated to complete the shape.

Shapes in Figures: A video lesson from Matholia that explores the 2-D shapes in other figures.

Describing and Naming Shapes: A video lesson from Matholia describing the properties of common flat (2D) shapes, including squares, rectangles, triangles, circles, semi-circles and quarter circles.

Who am I? A video lesson where children have to identify the 2-D shapes from their properties and pictures. The shapes at the beginning are those relevant to first and second classes and the latter shapes are more relevant to 3rd class up.

What am I? Read the clue on the card; do you know what shape is being described? These are printable but they could also be downloaded, read out and answered out loud, without having to print.

That Quiz Shapes: lots of different options here; start with “identify” and chose the shape names and level of difficulty to suit.

Pattern Blocks: Make numerous designs, pictures etc with these interactive pattern blocks. You can also choose a puzzle to complete.

Geoboard: Make lots of different shapes using this interactive geoboard, free from the Math Learning Centre.

2-D Shapes: A selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. Activity M1-M5 are all about 2-D shapes.

Math Games: a whole suit of geometry games, for all class levels; choose the skill you want to practice.

Digital Resources for Third to Sixth Classes

NB: Children in these classes may also enjoy the links for first and second classes, above

Maths is Fun: Background information on 2-D shapes as a part of geometry.

Matholia: Various video lessons from Matholia exploring the properties or characteristics of a rectangle, square, triangle, rhombus and trapezium.

Who am I? A video lesson where children have to identify the 2-D shapes from their properties and pictures. The shapes at the beginning are those relevant to first and second classes and the latter shapes are more relevant to 3rd class up.

What shape am I? This time you have to identify the shapes just from their properties. Make sure you guess before clicking on to see the answer!

That Quiz Shapes: lots of different options here; start with “identify” and chose the shape names and level of difficulty to suit.

That Quiz Triangles: lots of different options here; to identify different triangles, to calculate the measure of the angles, perimeter, area etc. Just chose the options and level of difficulty to suit.

2-D Shapes: Lots of useful information about 2-D shapes from BBC Skillswise, including a video highlighting 2-D shapes in the real world.

Pienado: A 2-D shape adventure game where you need to use 2-D shapes, in various positions, to plug gaps in a forcefield.

Classifying Triangles: a video which shows how all triangles are not the same.

IXL: A selection of geometry games from ixl.com. You can do a number of free quizzes each day without having a subscription.

Kangaroo Hop: Get your kangaroo to the finish line first by choosing the correct 2-D or 3-D shapes.

Khan Academy – Properties of Shapes (5th and 6th class): Watch this series of videos on triangles and quadrilaterals and answer the practice questions

I know it – Quadrilaterals: Interactive quiz

Khan Academy – Coordinates: (6th Class) Watch this series of videos and answer the practice questions

That Quiz Coordinates: (6th Class) From the options on the left hand side select identify/plot/both and quadrants I.

Polygon quiz: Name the polygons by dragging the names into the correct places.

Math Games: a whole suit of geometry games, for all class levels; choose the skill you want to practice.

Digging Deeper into … 2-D Shapes and 3-D Objects (infants to second class)

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes and 3-D objects, please check out the following posts:

Dear Family, your Operation Maths Guide to 2-D Shapes

Dear Family, your Operation Maths Guide to 3-D Objects

3-D shapes or 3-D objects?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe both 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is titled 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the 1999 Primary Maths curriculum.

So what first? 2-D or 3-D?

2-D and 3-D objects are very inter-related, to the point that there is often much debate about which of the topic should be taught first; 2-D shapes, 3-D objects or teach them both concurrently.

Since 2-D shapes are lacking the third dimension of depth or height that their 3-D relations possess, this makes them quite abstract as only flat, drawn/printed shapes are truly 2-D. Whereas, 3-D objects can be picked up, manipulated, used for constructions etc., making them much more suited to the concrete learning experiences that are essential in the early years. They are the objects that we find in the real-world. Thus, since 2-D shapes are only flat representations of the faces of 3-D objects, it could be argued that it would be more logical, and more in line with the concrete-pictorial-abstract (CPA) approach, to teach about 3-D objects before 2-D shapes.

On the other-hand, it could be argued that 2-D shapes should be taught first as it is likely than young children would be more familiar with them. For example, the vocabulary of 2-D shapes features more regularly in common speak than the vocabulary of 3-D objects. Many children will likely have encountered many 2-D shapes from picture books and patterns around their homes, etc. And so, it remains inconclusive as to which order of progression is most beneficial!

In the Operation Maths books, the children meet the specific topic of 2-D shapes prior to that of 3-D objects each school year. However, it is envisaged that by the time the children in the junior classes are formally engaging with 2-D shapes, they have already encountered and informally explored both 2-D shapes and 3-D objects via the monthly themes (laid out in the long-term and short-term plans of each TRB) and in the suggested Aistear play activities (detailed also in the TRB) of which, the Aistear theme of construction is particularly relevant.

Infant classes

Whether considering 2-D shapes or 3-D objects, the suggested progression within each topic is very similar:

• Undirected play
• Sorting and ordering activities
• Building and making (including making patterns)
• Identifying

Undirected play may include sand and water play, use of formal construction toys, constructing using “junk” or found materials; any activities that allow the children to handle and manipulate shapes and objects.
In the Operation Maths TRBs for junior and senior infants there are ample suggestions for suitable activities, under the headings of various themes. “Undirected play” does not imply that the teacher is superfluous to the process; rather while the children are the instigators, the teacher can play a vital role, observing the way in which the children interact with the materials, and asking the children to explain what they did, how they did it and why they did it that way. This can be a great way to assess the prior knowledge and language that the children may already have.

Sorting and ordering activities include the Early Mathematical Activities (EMA) used early on in the infant classes; thus it is likely that shapes and/or objects have already been used as part of these activities, for example sorting and matching according to colour, size etc; ordering according to length/height etc.

At this point, the children should also be prompted to sort the shapes and objects according to their respective properties as relevant and appropriate:

• Sort 2-D shapes according to the number of corners and the number and type of sides (straight, curved or both; sides that are different or the same).
• Sort 3-D objects according to those that roll/do not roll, slide/do not slide, build/do not build, are hollow/solid; as a development, according to the number of corners and the number and type of faces and edges (please see end of post for more information on faces and edges).
• The teacher can also isolate shapes/objects to create sets and then ask the children to identify the rule of the set: “What’s my rule?” (see image above). The children can also be encouraged to play the “What’s my rule?” game in groups.
• Isolate a particular shape/object in the room and ask the children to locate others that are the same/similar and make a set of like objects/shapes.
• The children may also be naming the shapes as part of these explorations; however this is not necessary as it is more important that they appreciate the similarities and difference between shapes, rather than identifying them.

Building and making with shapes/objects may have already been explored informally as part of the undirected play phase. The purpose now is to develop this into more formal teacher-directed tasks and activities:

• Build the tallest building/castle that you can. What objects did you use/not use and why?
• Dip a face of a 3-D object in paint and use it to print. Make a pattern using the prints. What do you notice?
• Try printing with different faces of the same 3-D object. Are the resulting prints the same or different?
• Push 3-D objects into sand/plasticine to make imprints. Or (if able) trace around the 3-D objects on paper to make designs.
• Use cut-out shapes, gummed shapes, tangrams and/or pattern blocks to make pictures and patterns.
• Use the shapes to cover surface of your book/mini-white board; which shapes did you use/not use and why?
• Combine two shapes/objects to make a new shape/object.

As part of the building and making activities the children may begin to realise how certain shapes/objects can be combined to become other shapes/objects. Similarly, through the reverse of these activities, and other shape cutting activities, the children should begin to realise that shapes can also be separated (partitioned) to reveal new shapes. This can include deconstructing 3-D objects to reveal their net. These activities can be revisited once the children can also name the shapes/objects, so as to arrive at certain understandings and become more accurate with mathematical language eg that two squares can make a rectangle; that, when using tangrams, two of the same size triangles can be rearranged to make a square, a larger triangle etc.

Identifying the specific shapes/objects evolves from the previous activities as the children begin to realise that it is the specific properties and attributes of a shape/object that defines it, eg all shapes with three corners (and three sides) are triangles, irrespective of their size or colour and irrespective of the measure of sides and corners (later to be referred to as angles). Activities which will serve to reinforce this include “Guess the shape/object” using descriptions (see below), guessing unseen shapes/objects from touch (eg in a feely bag), locating a specific shape from a collection using touch alone.
Through the experiences of printing and imprinting with the 3-D objects, it is also hoped that the children realise that the flat faces of 3-D objects are in fact 2-D shapes.

First and Second classes

The children in these classes will continue to sort, describe, compare and name shapes as done in infants, but to now also include new shapes and objects i.e. semi-circle (1st), oval and cone (2nd). They will continue to construct and make shapes, extending this to creating and drawing the shapes themselves.

They will further explore the combining and partitioning of 2-D shapes, and this understanding will extend to include the fractions of halves (1st) and quarters (2nd). The properties of 2-D shapes will be further explored
in second class via the stand units of symmetry, angles and area (i.e. tessellating 2-D shapes)

Q: How many faces on a cylinder? Three or two?
Traditionally, in Ireland, and in Irish textbooks, a cylinder was recorded as having three faces. However, this is not mathematically correct, as strictly speaking a face is flat, and a 2D shape (figure), so therefore a cylinder has in fact only two faces, (both circles), and one curved surface. And while it may be argued that a cylinder has a third face i.e. the rectangular shape you see when you disassemble the net of the 3-D object, in this disassembled state it is no longer a cylinder, since it can no longer roll, a specific property of all cylinders.
Another way to think about the faces of 3-D objects is to consider the number and shape of the resulting outlines of tracing around, or printing, each surface of the 3-D object. It is only possible to trace around the opposite ends/bases of the cylinder, since only these are flat, thus it has only two faces, both of which are circular in shape. Similarly, it is only possible to trace around one surface on a cone, which therefore means it has only one face (a circle) and one curved surface.
And how many edges on a cylinder? Officially none, as an edge is where two flat faces meet and the faces on a cylinder are on opposite sides and do not touch/meet. However, that leaves the problem of how to describe the place where each face meets the curved surface.  So in Operation Maths, as occurs typically in other primary texts in other countries, there is a distinction made between straight edges (which are in fact true edges) and curved edges (which strictly speaking are not edges).

Digging Deeper into … Spatial Awareness

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Spatial awareness…being able to describe the position of something/someone in relation to another using words and/or gestures, and being able to represent spaces and locations using models and/or drawings, may, at first glance, appear to have more in common with communication and geography, than with maths. However, the concepts of spatial awareness lay the foundations for all geometric thinking, be it at upper primary, secondary or an even higher level.

Essentially the children need to develop an understanding that:

• The spatial relationships between objects and places can be described and represented.
• These relationships may be viewed, described and represented differently depending on the perspective of the viewer (in particular, consider left and right).
• Developing the ability to mentally visualise the representations will enhance a person’s ability to picture how a shape will look when rotated when turned, flipped etc.

A synopsis of the curriculum objectives for infants to second class, state that the children should be enabled to:

• explore, discuss, develop and use the vocabulary of spatial relations (describing both position and direction/movement)
• explore closed shapes and open shapes and make body shapes
• give and follow simple directions (first and second class), including turning directions using half and quarter turns (second class only)
• explore and solve practical problems (first and second class)

In the case of the practical problems, this could include completing a jigsaw or a tangram puzzle, using mazes, grids, board games and or exploring basic coding eg via coding programs and apps, such as Lightbot, and more hand-on devices such as BeeBots.

Moving through space

Since spatial awareness requires an understanding of using  space and moving through space, the majority of the activities should be active ones, where the children are moving around. This is where the suggested activities in the Operation Maths Teachers Resource Book (TRB) become extremely useful, such as the examples below.

Much of the language development in this strand unit can be reinforced via activities in PE (Orienteering) and Geography (mapping).

Digital Resources

While activities incorporating physical movement are preferable, the Operation Maths digital resources on edcolearning, provide a worthwhile alternative and add variety. The Ready to Go activities below, as the phrase says are “exactly what they say on the tin”; the teacher need only click on the relevant icon in the digital version of the pupil’s book to open the activity, and the accompanying suggested questions are quickly viewable along the side menu. A full description of the activity, including the questions, is also given in the TRB.

Digging Deeper into … Symmetry (2nd to 4th)

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of symmetry, please check out the following post: Dear Family, your Operation Maths Guide to Symmetry.

Symmetry is officially a strand unit for second to fourth classes, although it also features as a content objective in 2-D shapes for fifth and sixth class where the children “classify 2-D shapes according to their lines of symmetry”.

While there are different types of symmetry, the curriculum specifies line symmetry, also known as mirror symmetry, reflective or reflection symmetry.

In Operation Maths, this chapter is placed after 2-D shapes, so that the children can identify symmetry in the shapes that they have previously encountered, and, in third  and fourth class, it is placed after Lines and Angles so that they can use their knowledge of different line types when describing the lines of symmetry.

Concrete exploration

To complete or create symmetrical patterns, requires the children being able to visualise the mirror image of the given arrangement/image. But children cannot visualise what they have not experienced. Thus to experience symmetry the children must:

• be made aware of examples of symmetry all around them, and locate examples themselves e.g. flowers, leaves, objects at home and at school, numbers and letters of the alphabet.
• be afforded ample opportunities to use real mirrors to explore symmetry. The type of child-safe mirrors that are often used in science investigations (eg in the strand unit of light) are ideal for this purpose.

Using mirrors allows the children the opportunity to observe symmetry and to check the accuracy of their completed patterns.  When using mirrors:

• Try to have enough mirrors for one between two (the child-safe mirrors can often be cut into smaller sizes, 10cm x 7cm approx is big enough), or if supply is limited the mirror exploration could be incorporated as a station in a station/team teaching maths lesson.
• Initially, allow the children free exploration and then, when suitable, guide it towards a purpose using questioning:
• What letters or numbers look the same in the mirror? What shapes or images in the environment look the same in the mirror?
• Can you put the mirror along the middle of any shapes and numbers so that they look complete? Does this work with any shapes or images from the environment? Don’t specify “middle” as being horizontal or vertical, and then see if the children realise that, on some figures, there is more than one than one way that the mirror can be placed.
• At this point you could use this as the introduction to a separate and distinct What do you notice? What do you wonder? activity, and use the children’s wonder questions to guide the course of the rest of the lesson.
• Explain that, on the symmetrical figures, the position of the mirror, is referred to as the line of symmetry. Then ask the children to use the mirrors to identify/draw the line of symmetry on the figures or mark the line of symmetry first (more challenging) and then check using the mirror.
• Using the mirrors the children can create and check symmetrical patterns using cubes, counters, objects etc. One child can create a pattern that their partner has to complete symmetrically. Since children often incorrectly replicate the pattern (eg as done in the first image below) rather than reverse it, the mirror can show them their error (as used in second image below). Encourage the children to realise that whatever is closest to the mirror/line of symmetry on one side will also be closest to the mirror on the other side.

• The children could then progress to creating symmetrical arrangements of more than one row. The Operation Maths twenty frames (free with Operation Maths 1 and 2) can be very useful for this (see below). Again the children should be encouraged to recognise that the colour and type of object/figure that is closest to the line of symmetry on one side should also be closest to the line of symmetry on the other side.

When the children have had sufficient experience with actual mirrors they should progress to completing activities without them, although they could always be returned to again if needs arose.

Digging Deeper into … 3-D Objects

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 3-D objects, please check out the following post: Dear Family, your Operation Maths Guide to 3-D Objects

The first obvious question to be addressed is why this topic is called 3-D objects in Operation Maths, when most other textbooks, and even the curriculum refer to this topic as 3-D shapes?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is called 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the curriculum.

CPA

As usual, this topic is explored using a CPA approach, where the initial focus is on the exploration of 3-D objects and the identification of similarly shapes objects from the child’s environments. And, in a similar way to the teaching of 2D shapes, while the 3-D objects will be identified by name, the greater focus should be on their properties, as appropriate to each class level e.g.

• identifying whether the objects roll, stack, or slide
• relating the properties of each object to its purpose and use in the environment
• recording, sorting and comparing according to the number and shape of faces and curved surfaces
• recording, sorting and comparing according to the number of edges and vertices (corners)

It is important that the children discover the properties of the 3-D objects by hands-on investigations and by classifying and sorting. Sorting activities help develop the children’s communication, observation, reasoning and categorising skills, and thus will help to develop a conceptual understanding of the objects. In the senior classes using online interactive sorting activities based on Venn diagrams or Carroll diagrams can be very useful. Asking children to bring in examples of 3-D objects from home will help them to become aware of 3-D objects in their environment.

Faces, curved surfaces and edges

How many faces on a cylinder? Three or two?

Traditionally, in Ireland, and in Irish textbooks, a cylinder was recorded as having three faces. However, this is not mathematically correct, as strictly speaking a face is flat, and a 2D shape (figure), so therefore a cylinder has in fact only two faces, (both circles), and one curved surface. And while some may argue that a cylinder has a third face i.e. the rectangular shape you see when you disassemble the net of the 3-D object, in this disassembled state it is no longer a cylinder, since it can no longer roll, a specific property of all cylinders.

Another way to think about the faces of 3-D objects is to consider the number and shape of the resulting outlines of tracing around each surface of the 3-D object. It is only possible to trace around the opposite ends/bases of the cylinder, since only these are flat, thus it has only two faces, both of which are circular in shape. Similarly, it is only possible to trace around one surface on a cone, which therefore means it has only one face (a circle) and one curved surface.

And how many edges on a cylinder? Officially none, as an edge is where two flat faces meet and the faces on a cylinder are on opposite sides and do not touch/meet. However, that leaves the problem of how to describe the place where each face meets the curved surface.  So in Operation Maths, as occurs typically in other primary texts in other countries, there is a distinction made between straight edges (which are in fact true edges) and curved edges (which strictly speaking are not edges).

In 5th and 6th class, it is important that the children realise that only some of the 3D objects are also polyhedra, and that only five of these are categorised as platonic solids.

Digging deeper into … the Circle (5th & 6th)

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While the circle, as a topic, is not a specific strand unit in itself, it traditionally has been dealt with separately, and more in depth, in 5th and 6th class. It is in these classes that the children begin to explore the circle as something more than just a 2D shape, and rather as a shape which has specific parts that can be named (eg diameter, radius, etc) and specific properties that can be explored. Fifth class is also the first time that the children have encountered degrees as a measurement of rotation (they first meet degrees in the strand unit of lines and angles). Fifth class is also the first occasion when the children would be required to use degrees to construct pie-charts in the strand unit of representing and interpreting data, for which the children require knowledge of constructing circles and dividing them into sectors prior to creating pie charts.

Thus, in Operation Maths, this chapter is placed after the chapters of Lines and Angles and 2-D Shapes and before the chapter of Data.

CPA Approach

As is typical in Operation Maths, A CPA approach is taken to this topic where the emphasis at the introductory stages is on the children exploring and examining circles in their environment and then using the manipulation of cut-out circles to identify the parts of circle and label them using the correct mathematical terminology. Indeed the children themselves can be used to model a circle, as outlined below. This type of activity will also greatly suit the kinaesthetic learners.

When looking for circles in the environment, it is also important to accurately identify circles, which are examples of 2D shapes, as distinct from cylinders, which are examples of 3D objects. So, if a child suggests that a coin is an example of a circle, emphasise that the face of a coin is indeed a circle, but that the coin itself is a cylinder.

Integrating maths and literacy

When introducing this new terminology, ask the children to suggest examples of words with similar prefixes/rootwords so as to foster connections and deepen meaning eg diameter coming from dia meaning across, through and thus connected with diagonal, diaphragm, dialogue; radius as being related to radiate, radio, radar (originating from a central point and moving outwards), etc. Etymology websites, such as Etymonline can be very useful to research and collect related words.

Measuring and constructing circles

The children should be provided with ample opportunities to measure the radius and diameter of circles of various sizes and, in doing so, be guided to discover for themselves that the measure of the diameter is twice the radius.

In a similar way, in 6th class, through comparing the measurement of the circumference and diameter of various circles, it is hoped that the children realise that the circumference of a circle is always just over three times the measure of the diameter. They can explore this in a very concrete way by measuring the circumference of various circles using string/wool and then cutting the string into lengths that equal the diameter, as show in the image below from K-5 Math Teaching Resources

From K-5 Math Teaching Resources

Again before constructing circles using a compass, the children should be asked to suggest ways to draw circles that they may have used previously and to identify the pros and cons of these methods. When ready to use a compass, it can be a good idea to use a video to demonstrate, such as the ones below.

Circles in art

Once the children have mastered the basics of constructing a circle, they could be encouraged to look and respond to circles in art, for example the work of Kandinsky, Anwar Jalal Shemza or the multitude of circle themed pieces available to view on the internet (just search google images for circles in art). The children could even look and respond to crop circles (in particular the activities of John Lundberg) or the use of circles in famous brand logos. For more ideas, check out this Pinterest board which includes circle themed art lessons. Using circles in such a way provides purposeful opportunities to use and reinforce the specific circle terminology e.g. diameter, circumference, arc, etc.

Shape and Space Manual from PDST, p.154-157

Digging Deeper into … 2D Shapes (3rd – 6th)

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For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes, please check out the following post: Dear Family, your Operation Maths Guide to 2-D Shapes

Overview of 2D-Shapes:

The following are the new 2D shapes, to which the children are formally introduced, in the senior end of primary school:

• 3rd class: hexagon
• 4th class:  Equilateral, isosceles and scalene triangles; rhombus & parallelogram; pentagon and octagon
• 5th class: Polygons, quadrilaterals, trapezium
• 6th class: Kite*

* while the kite is not specified on the Primary Mathematics Curriculum (1999) for sixth class, it has been included in Operation Maths 6 as it features on the curricula for 5th/6th grade in many other countries.

As with every topic in Operation Maths, a CPA approach is also recommended for 2D shapes:

Concrete: Using concrete geometric shapes for classifying and sorting; identifying examples of 2D shapes and tessellations in the environment.
Pictorial: Tracing around shape templates to make reproductions that can be manipulated, folded, partitioned and combined; using lollipop sticks, geostrips or geoboards to create representations of 2D shapes; using the Operation Maths Scratch lessons accessible on edcolearning.ie to draw various shapes.
Abstract: Answering shape questions with no visual references/supports; suggesting the number of lines of symmetry on a shape without folding or drawing to explore the same; identifying the resulting shape when a given shape is rotated, flipped etc.

Properties of 2D Shapes:

Don’t let the list above, of 2D shapes by class, fool you; it shouldn’t create an incorrect impression that the primary focus is on identifying shapes, or that we should look at one type of 2D shape exclusive of others. Rather, the focus should be on the children examining the properties of each 2-D shape, describing it according to these properties and contrasting it with, and comparing it to, the properties of other shapes, rather than on just naming the shape. For example, what makes a square a square? How is a square similar to, or different from, a rectangle? Could an argument be made to say a square is also a rectangle? Could an argument be made to say a rectangle is also a square?

Therefore, any new 2D shapes that the children encounter should be compared to the 2D shapes with which they are already familiar. And, as the 2D shapes chapter in Operation Maths always follows on from the topic of Lines and Angles, when exploring properties, reference should also be made to the number and type of angles within the shape, the number and types of sides (parallel, perpendicular etc) and whether the shape is regular or irregular.

Common misconceptions:

Categorising 2D shapes  separately: As mentioned previously, children often don’t recognise a square as a type of rectangle, a rectangle as a type of parallelogram, a rhombus as a type of parallelogram etc. This can often be the case if the children are focused primarily on naming the shape and then compartmentalising it in a category, as opposed to examining its properties and exploring how it may have proprieties in common with other shapes.

It can be useful here to display 2D shapes to the class using a subgroup structure (like this one here) so that the children can appreciate how, for example, a square is also a rectangle, is also a parallelogram, is also a quadrilateral, is also a polygon.

Constancy of shapes: Many children don’t recognise that a  shape remains constant, irrespective of its placement in space. In particular, a square sitting on its vertex is often incorrectly labelled as a diamond. The children should be encouraged to draw or trace around shapes on their MWBs and then rotate the shape in order to appreciate its constancy.

Regularity: Children may not recognise a five-sided figure with sides of same length as being a regular shape. It is as if the terminology “regular” implies to them that the shape should be common-place i.e. regularly occurring. For this same reason, a child will often say a rectangle and a circle are regular shapes, given their familiarity with these shapes from the junior classes, even though they are officially classified as irregular shapes. Challenging this misconception will require plenty of sorting activities where shapes are classified as regular or irregular (see Ready to Go activities below).

When creating a class display of shapes eg rectangles, triangles, etc., instead of using just one qualifying shape to illustrate the term, use many and use varied ones. Enlist the help of the class: “I want to make a display of triangles/parallelograms but I want the triangles/parallelograms to all be different. Can you draw and cut some out for me?” Such an activity would quickly reveal those who appreciate the required properties for each shape and those who don’t. Remember to also position the shapes in various ways so as to reinforce that the shape remains constant, irrespective of placement.

Identifying 3D objects as 2D shapes: This is a very problematic area. It often happens that when asked to find a circle in the environment, a child suggests a ball (a sphere) or a cube might be suggested as a square. When asking to identify 2D shapes in the classroom or at home, we must be careful how we respond to the answers so as not to reinforce these misconceptions. For example, if a child suggests that the door is a rectangle, when it is in fact a cuboid, emphasise that a part of the door is rectangular eg

• Which part of the door is like a rectangle?
• Are there any other parts of the door that are like a rectangle?
• Can you see any other rectangles on the outside of the door? How many?
• Are they all the same or different?

Asking the children to draw around solid 3D objects in order to produce flat 2D shapes can also be useful here.

Coordinates (6th class)

The concept of plotting and reading coordinates is introduced in 6th Class. There are plenty of examples of coordinates in the children’s environment, e.g. map reading, car parks and board games such as chess and Battleship. Allow the children to practise reading coordinates on maps and on board games, then progress to using two digit coordinates in maths. Make sure they first read the horizontal coordinate and then the vertical coordinate.

Operation Maths Digital Resources:

Don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie . These include:

Ready to go Activities: based on the Sorting eManipulative, these enable the various shapes to be sorted according to class-appropriate criteria, or enable tessellating patterns to be made. The Ready to go activities all have suggested questions inbuilt on the left-hand side of the screen that the teacher can just click to reveal and hide. Remember, when sorting, the focus should be on the properties of the shapes not their names; that said, you can also ask the children to identify the shapes, if known, as an extra dimension to the activity.

Create activities: (all classes) again using the Sorting eManipulative, these are less structured that their Ready to go counterparts. Instead, the teacher should click on the yellow “Create new example” button on the bottom of the screen, and then use the sorting eManipulative to explore the shapes as they see fit. The teacher can use a previous Ready to go activity to inspire the create activity or come up with a completely different activity of their own using the almost limitless possibilities of the sorting eManipulative.

Write-Hide-Show videos: These explore tessellations (3rd class) and different types of triangles (5th class). They encourage the children to look and respond to the questions by answering orally or on their MWBs.

Maths Around Us video (6th class): which examines different types of triangles from the environment.

Scratch-based programming lessons with instructions on how to draw 2D shapes  (3rd class) and hexagons (3rd, 4th, 5th classes), scalene triangle, pentagon and octagon (4th class), different types of triangles (5th class) and plot 2D shapes on a grid (6th class).

Digging Deeper into … Lines and Angles

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of place value, please check out the following post: Dear Family, your Operation Maths Guide to Lines and Angles

Overview:

As can be seen from the overview table below, this topic is initially introduced to the children in 2nd class via “turns” and “square corners” and then develops with increasing complexity in 3rd- 6th classes.

 2nd 3rd 4th 5th 6th Lines vertical horizontal parallel perpendicular oblique Angles Full, half, quarter turns, square corner Right angle Greater/less than a right angle Acute angle Obtuse angle Straight angle Reflex angle Measuring and constructing in degrees Sum of angles in triangle  = 180° Sum of angles in quadrilateral  = 360°

As with every topic in Operation Maths, a CPA approach is also recommended for lines and angles:

• Concrete: allow sufficient time for the children to explore making turns, lines and angles with suitable concrete materials (e.g. the children themselves, lollipop sticks, straws, geo strips, construction materials, real-life examples from the school and home environment)
• Pictorial: activities where the focus is on drawing angles or lines on paper, MWBs etc
• Abstract: the final stage, where the focus is primarily on numbers,  digits  and or letters to represent variables eg given the measure of two angles in a triangle calculate the third angle.

Lines:

Through exploration and activities, it is important that the children realise that:

• A line can be classified and identified according to its position and its relationship to another line.
• A single line can be horizontal, vertical or oblique but a single line cannot on its own be parallel or perpendicular; there must be two or more lines.
• Parallel lines do not all have to be the same length to be parallel.
• Parallel lines do not have to be horizontal or vertical, they can also be oblique.
• Perpendicular lines do not have to have a horizontal and vertical line, (again they can be oblique) but there must be at least one right angle where the two lines meet.

Lines can be drawn on the Operation Maths MWBs and then rotated to reinforce this point.
It is also worth noting that in maths, when we use the word “line”, it should be assumed that this is always straight; only if the word curved is given should it be assumed otherwise.

Angles

In order for the children to recognise angles in terms of rotation, it is preferable initially, for the children to investigate the angles in their environment that are dynamic, (where the angle can be easily made bigger or smaller by increasing or decreasing the distance between the two lines) e.g. a door opening and closing, a scissors cutting paper, the angles made by the hands of a clock. The children can then proceed to examine static angles (where the angle is fixed) e.g. in 2-D shapes or 3D objects.

Operation Maths, Pupils’ Book 3: on the digital book, click on the icon on the bottom right to access a “Ready to go” activity.

In second class (and revised at the start of third class), the concept of rotation of an angle is taught through the terms quarter-turn, half-turn and full turn. Ideally, this should be introduced concretely by getting the children themselves to do half-turns and quarter-turns, and to turn in clockwise and anticlockwise directions:

• In the classroom, the children start facing the board/front of room and make half/full/quarter turns to left/right as directed by the teacher.
• Repeat, but this time with different starting points
• Repeat, but this time after the teacher gives the directions the children must say where they will be facing, before they do the actual turn. The children could also record their predictions quickly on their Operation Maths MWBs

The children will be also be asked to identify 90º angles as square corners (2nd class) or right angles (3rd class). This will also be reinforced as part of the 2D shapes chapter. The children can be asked how they might decide if a corner/vertex of a shape is a square corner/right angle. Prior to the introduction of the protractor, something as simple as a corner torn from a piece of paper would suffice as an instrument with which to measure these angles.

The children should be enabled to classify angles according to the criteria appropriate to their class level (see table above). In particular, the ability to identify angles as acute (or less than a right angle), obtuse (or greater than a right angle) or reflex will greatly help the children, to later, estimate the measure of the angle in degrees, and to accurately measure and construct angles when they encounter this in 5th and 6th classes.

Operation Maths, Pupils’ Book 5

It is important to constantly reinforce the children’s understanding of what an angle actually is, i.e. an amount of turn and that this can be represented by two adjoining lines, one showing the starting position, the other showing the point after the turn. Return to concrete examples if necessary; the children stretch out two arms in front and, leaving one arm in original position, they move other arm a certain amount (90 º, 180º etc). This could also be repeated using geostrips, connected at one end using a brass clip, so as to be able to move one of the ‘arms’. Such concrete experiences also link well to measuring using a protractor; the original arm is the ‘base’ line.

Measuring and constructing angles (5th & 6th classes)

Operation Maths, Pupils’ Book 5

Using degrees to describe angles is introduced in 5th class, which develops to include measuring and constructing angles using degrees. This necessitates the use of a protractor for the first time, which in itself can lead to difficulties. The child may be unsure where exactly to place the protractor; this can come from a lack of understanding of what an angle actually is and where the angle actually is. Also, a child can be uncertain of which scale to use to measure the size of the angle eg for an acute angle measuring 45º, the child writes down 135º.

To reduce the likelihood of this arising you can ask some/all of the following questions:

• What important tips would you give to a person about using a protractor?
• How do you know which scale to use on the protractor?
• What type of angle is this? How do you know? (To save time, they can write A, O or Re for Acute, Obtuse or Reflex).
• Estimate the measure of the angle to the nearest 10º. Is your estimate/measurement sensible? Why?
• How can you use what you know about acute and obtuse angles to check your measurement?

You can also watch some of the tutorial videos for using a protractor on the internet, such as the one below, for example. These videos can be a very visual way of demonstrating this skill. See also the list of digital resources for 5th and 6th classes on the following post: Dear Family, your Operation Maths Guide to Lines and Angles.

Angles is another area where it is important for the child to check the reasonableness of the answer. First, the child needs to identify whether the angle is acute or obtuse. Then, if measuring an acute angle, the measurement must be less than 90º. If it isn’t, then the correct scale wasn’t used.

Lines and Angles all around us

It is a given that lines and angles are all around us, although children may often be oblivious to the examples! Again, appropriate to each class level, the children should be encouraged to identify different types of lines and angles in their classroom, school and home. Enrich your own classroom space with lines and angles by labeling the line types in the room and the measure of angles of the open door. Make it personal by relating this topic to the children themselves and to the geometry in their names. Incorporate lines and angles into your visual art lessons (see also image below). Operation Maths 4 and 5 users can show the Maths Around Us video to their class, accessible on www.edcolearning.ie. For more ideas, check out this Lines & Angles board on Pinterest.

Operation Maths, Pupils’ Book 4